A cylindrical tube moves at a constant velocity downward wit

A cylindrical tube moves at a constant velocity downward with the tube axis inclined at an angle from the vertical through a large body of stationary incompressible fluid.

A. Assuming the flow inside the tube to be laminar, derive an expression for the steady-state velocity profile in the tube starting from the Navier-Stokes equation. State all assumptions clearly.

b. A cylindrical tube with an internal diameter of 1 cm moves with a constant velocity of 10 cm/s downward with the tube axis inclined at an angle of 30° from the vertical through a large body of stationary water. Assume that the density of water is 1000 kg/m³ and its viscosity is 0.001 Pa.s. Use the expression derived above to plot the velocity profile in the tube.

c. Verify if the laminar flow assumption is valid for the conditions in part b.

Solution

According to Newton\'s second law of motion

F_x =m*a_x

in the fluid, following forces are present

1)Fg,gravity force

2)F_p,pressure force

3)F_v,force due to viscocity

4)F_t, force due to turbulence

5)F_c, force due to compressibility

from th eabove equation

the net force F_x =(F_g)_x +(F_p)_x +(F_v)_x + (F_t)_x

the above eq is navier stroke\'s eq

internal dia of tube D=1 cm

velocity V= 10 cm/s

angle =30 degrees

density of water 1000 kg/m³

viscocity =0.001 pa.s

according to eq of continuity

Q=AV

A=